1. Technical Field
The present disclosure relates to signal reconstruction and, more specifically, to a system and method for signal reconstruction from incomplete data.
2. Discussion of Related Art
Electronic signals are commonly represented by digital data. These electronic signals may represent audio data, image data and a wide range of other forms of data. Often, the digital data that represents the electronic signal only contains partial information. This occurs, to some extent, when the process of digitization samples only at regular predetermined intervals, rather than continuously. The resulting digital signal therefore may not include all of the information necessary to fully reproduce the original electronic signal. This lack of complete information may be exacerbated when the digital data is subjected to lossy digital compression techniques that tend to substantially reduce file size by excluding certain data that is calculated to have a reduced impact on signal quality. Incomplete reception of transmitted digital data may also result in the obtaining of partial information.
Reconstruction is the process of reproducing electronic signals from digital data. Reconstruction may therefore be used to reproduce sound from a sound file, to reproduce an image from an image file, or to reproduce another form of signal from digital data. For example, in the field of medical imaging, reconstruction may be used to generate tomography data from detected x-rays, in the case of computed tomography (CT), or from detected magnetic fields, in the case of magnetic resonance imaging (MRI). In such cases, the collected digital data may also be incomplete. Incomplete data collection may, for example, be the result of an intentional collection of only partial data for the purposes of speeding up the process of image data acquisition. Partial data may also be collected in order to lower a radiation dose that a patient is exposed to during CT acquisition.
Thus, it is often necessary to perform reconstruction on incomplete data. For example, sound waves may be reconstructed from incomplete digital sound data and magnetic tomography images may be reconstructed from incomplete digital magnetic field data. In reconstruction, missing data may be approximated as best as possible based on the characteristics of the available data.
There are many approaches for approximating missing data based on available data. One such approach starts with the assumption that signals, such as those representing sound and images, tend to have smooth characteristics that can be expressed in terms of zero-value gradients or zero-value results from another form of transform. Such transforms may be calculated, for example, by a gradient transform, a wavelet transform or another suitable approach. Thus, the missing data may be approximated as that data which provides for the complete signal, a maximum number of zero-value gradients, or more precisely, a minimum number of non-zero value gradients. Such approaches to reconstruction of signals from incomplete data are known as “sparse reconstruction” as they seek to maximize “sparsity” which is defined conventionally as the characteristic of having zero-values.
However, in practice, minimizing non-zero gradients in reconstructed data can be quite difficult and may not be effectively computable within an acceptable amount of time.